Answer to Question #176264 in Algebra for ankit kumar

Question

Give an example, with justification, to show why 0 a 1 < i.

Answer 1

The definition of "Factorial" according to Wikipedia says;

It is a non-negative integer n, denoted by n!, is the product of all positive integer less than or equal to n.

$n!=n×(n−1)×(n−2)×(n−3)×...3×2×1$

However the recursive definition of factorial is of more use in this proof.

$n!={1n×(n−1)}!n=1n≥0$

Recursive definition of Factorial leads to one interesting way of expressing factorial numbers.

$n!=\dfrac{(n+1)!}{(n+1)}$

This is valid since, as we expand (n+1)! from recursive definition, we can cancel (n+1) term from both numerator and denominator to get n!. Or we can even calculate factorial in numerator and then evaluate the division.

For example,

$5!=\dfrac{6!}{6}= \dfrac{720}6 \\$

$4!=\dfrac{5!}{5}=\dfrac{120}5$

$3!=\dfrac{4!}{4}=\dfrac{24}4$

$2!=\dfrac{3!}{3}=\dfrac63$

$1!=\dfrac{2!}{2}=\dfrac22$

In a similar way, if we try to express 0! we get

$0!=\dfrac{1!}1=1$

And this ends our proof that $0!=1$ .

This proof is one of many ways, where 0! leads to 1. But this one is quite explanatory in itself.